2,215 research outputs found
A new numerical scheme for solving system of Volterra integro-differential equation
In this article, we apply Genocchi polynomials to solve numerically a system of Volterra integro-differential equations. This is done by approximating functions using Genocchi polynomials and derivatives using Genocchi polynomials operational matrix of integer order derivative. Com-bining approximation with collocation method, the problem is reduced to a system of algebraic equations in terms of Genocchi coefficients of the unknown functions. By solving the Genocchi coefficients, we obtain good approximate functions of the exact solutions of the system. A few numerical examples show that our proposed Genocchi polynomials method achieves better accu-racy compared to some other existing methods
Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations
In this paper, we use a numerical method that involves hybrid and block-pulse
functions to approximate solutions of systems of a class of Fredholm and
Volterra integro-differential equations. The key point is to derive a new
approximation for the derivatives of the solutions and then reduce the
integro-differential equation to a system of algebraic equations that can be
solved using classical methods. Some numerical examples are dedicated for
showing efficiency and validity of the method that we introduce
On Some Operators Involving Hadamard Derivatives
In this paper we introduce a novel Mittag--Leffler-type function and study
its properties in relation to some integro-differential operators involving
Hadamard fractional derivatives or Hyper-Bessel-type operators. We discuss then
the utility of these results to solve some integro-differential equations
involving these operators by means of operational methods. We show the
advantage of our approach through some examples. Among these, an application to
a modified Lamb--Bateman integral equation is presented
A new operational matrix based on Bernoulli polynomials
In this research, the Bernoulli polynomials are introduced. The properties of
these polynomials are employed to construct the operational matrices of
integration together with the derivative and product. These properties are then
utilized to transform the differential equation to a matrix equation which
corresponds to a system of algebraic equations with unknown Bernoulli
coefficients. This method can be used for many problems such as differential
equations, integral equations and so on. Numerical examples show the method is
computationally simple and also illustrate the efficiency and accuracy of the
method
Analytic solutions of fractional differential equations by operational methods
We describe a general operational method that can be used in the analysis of
fractional initial and boundary value problems with additional analytic
conditions. As an example, we derive analytic solutions of some fractional
generalisation of differential equations of mathematical physics. Fractionality
is obtained by substituting the ordinary integer-order derivative with the
Caputo fractional derivative. Furthermore, operational relations between
ordinary and fractional differentiation are shown and discussed in detail.
Finally, a last example concerns the application of the method to the study of
a fractional Poisson process
A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations
In this paper we introduce a numerical method for solving nonlinear Volterra
integro-differential equations. In the first step, we apply implicit trapezium
rule to discretize the integral in given equation. Further, the Daftardar-Gejji
and Jafari technique (DJM) is used to find the unknown term on the right side.
We derive existence-uniqueness theorem for such equations by using Lipschitz
condition. We further present the error, convergence, stability and bifurcation
analysis of the proposed method. We solve various types of equations using this
method and compare the error with other numerical methods. It is observed that
our method is more efficient than other numerical methods
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